Collapsing Riemannian Metrics to Sub-Riemannian and the Geometry of Hypersurfaces in Carnot Groups

Given a Carnot group with step two G, we study the limit as e → 0 of a family of left-invariant Riemannian metrics g e on G. In such metrics, neither the Ricci tensor nor the sectional curvatures are bounded from below. Nonetheless, our main result shows that the Riemannian first and second variation formulas in the g e-metrics converge to the corresponding sub-Riemannian ones. This testifies of an intrinsic stability of some deli- cate cancellation processes and makes the method of collapsing Riemannian metrics a possible tool for attacking various fundamental open questions in sub-Riemannian geometry. Finally, we mention that the restriction to groups of step two is purely for ease of exposition and that the same method can be applied to Carnot groups of arbitrary step.